The Annals of Statistics

An Unbalanced Jackknife

Rupert G. Miller

Full-text: Open access

Abstract

It is proved that the jackknife estimate $\tilde{\theta} = n\hat{\theta} - (n - 1)(\sum \hat{\theta}_{-i}/n)$ of a function $\theta = f(\beta)$ of the regression parameters in a general linear model $\mathbf{Y} = \mathbf{X\beta} + \mathbf{e}$ is asymptotically normally distributed under conditions that do not require $\mathbf{e}$ to be normally distributed. The jackknife is applied by deleting in succession each row of the $\mathbf{X}$ matrix and $\mathbf{Y}$ vector in order to compute $\hat{\mathbf{\beta}}_{-i}$, which is the least squares estimate with the $i$th row deleted, and $\hat{\theta}_{-i} = f(\hat\mathbf{\beta}_{-i})$. The standard error of the pseudo-values $\tilde{\theta}_i = n\hat{\theta} - (n - 1)\hat{\theta}_{-i}$ is a consistent estimate of the asymptotic standard deviation of $\tilde{\theta}$ under similar conditions. Generalizations and applications are discussed.

Article information

Source
Ann. Statist. Volume 2, Number 5 (1974), 880-891.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342811

Digital Object Identifier
doi:10.1214/aos/1176342811

Mathematical Reviews number (MathSciNet)
MR356353

Zentralblatt MATH identifier
0289.62042

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory

Keywords
15 35 Jackknife pseudo-value general linear model multiple regression asymptotic normality

Citation

Miller, Rupert G. An Unbalanced Jackknife. Ann. Statist. 2 (1974), no. 5, 880--891. doi:10.1214/aos/1176342811. https://projecteuclid.org/euclid.aos/1176342811


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